Simple Roots Research Articles

Efficient Multiplicative Calculus-Based Iterative Scheme for Nonlinear Engineering Applications

It is essential to solve nonlinear equations in engineering, where accuracy and precision are critical. In this paper, a novel family of iterative methods for finding the simple roots of nonlinear equations based on multiplicative calculus is introduced. Based on theoretical research, a novel family of simple root-finding schemes based on multiplicative calculus has been devised, with a convergence order of seven. The symmetry in the pie graph of the convergence–divergence areas demonstrates that the method is stable and consistent when dealing with nonlinear engineering problems. An extensive examination of the numerical results of the engineering applications is presented in order to assess the effectiveness, stability, and consistency of the recently established method in comparison to current methods. The analysis includes the total number of functions and derivative evaluations per iteration, elapsed time, residual errors, local computational order of convergence, and error graphs, which demonstrate our method’s better convergence behavior when compared to other approaches.

Rogue wave patterns associated with Adler–Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation

AbstractIn this work, we analyze the asymptotic behaviors of high‐order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including modified claw‐like, one triple root (OTR)‐type, modified OTR‐type, two triple roots (TTR)‐type, semimodified TTR‐type, and modified TTR‐type patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler–Moser polynomials with multiple roots. At the positions in the ‐plane corresponding to simple roots of the Adler–Moser polynomials, these high‐order rogue wave patterns asymptotically approach first‐order rogue waves. At the positions in the ‐plane corresponding to multiple roots of the Adler–Moser polynomials, these rogue wave patterns asymptotically tend toward lower‐order fundamental rogue waves, dispersed first‐order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler–Moser polynomials with new free parameters, such as the Yablonskii–Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower‐order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.

A NEW METHOD FOR FINDING A SIMPLE ROOT

A NEW METHOD FOR FINDING A SIMPLE ROOT

Affine Standard Lyndon Words: A-Type

Abstract We generalize an algorithm of Leclerc [ 6] describing explicitly the bijection of Lalonde–Ram [ 5] from finite to affine Lie algebras. In type $A_{n}^{(1)}$, we compute all affine standard Lyndon words for any order of the simple roots and establish some properties of the induced orders on the positive affine roots.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

A generalized approach for pricing american options under a regime-switching model

Abstract Regime-switching models gained their popularity over the past decade because of their distinctive advantage of modelling different financial market statuses in a discrete manner rather than a continuous manner as in stochastic volatility models. When they are used in option pricing, the clear advantage is that they enable a larger parameter space for models to be calibrated for a specific market dynamics and thus allow a better quantitative risk management in terms of utilizing financial derivatives. However, when they are used to price American-style financial derivatives, a large number of economic statuses result in a demand for improved computational efficiency. This paper provides a new algorithm of high computational efficiency supplemented with a theorem that pre-analyzes the associated matrices and their eigenvalues, without concern about the possibility of duplicated roots being mistakenly identified as simple roots due to rounding errors or the presence of two extremely closely positioned simple roots within the original system.

An atlas of Brachypodium distachyon lateral root development.

The root system of plants is a vital part for successful development and adaptation to different soil types and environments. A major determinant of the shape of a plant root system is the formation of lateral roots, allowing for expansion of the root system. Arabidopsis thaliana, with its simple root anatomy, has been extensively studied to reveal the genetic program underlying root branching. However, to get a more general understanding of lateral root development, comparative studies in species with a more complex root anatomy are required. Here, by combining optimized clearing methods and histology, we describe an atlas of lateral root development in Brachypodium distachyon, a wild, temperate grass species. We show that lateral roots initiate from enlarged phloem pole pericycle cells and that the overlying endodermis reactivates its cell cycle and eventually forms the root cap. In addition, auxin signaling reported by the DR5 reporter was not detected in the phloem pole pericycle cells or young primordia. In contrast, auxin signaling was activated in the overlying cortical cell layers, including the exodermis. Thus, Brachypodium is a valuable model to investigate how signaling pathways and cellular responses have been repurposed to facilitate lateral root organogenesis.

Exploring Weight Functions: A Family of One-Point Methods for Multiple Roots

In this paper we introduce a family of one-point methods for multiple roots using weight functions. We analyze the necessary conditions for the weight function to achieve third-order accuracy, and then present a sequence of rational weight functions for a complex quadratic function. The resulting members of the family exhibit Julia sets that align with the perpendicular bisector of the roots. We compare selected members with existing methods based on basin of attraction analysis, demonstrating their competitive performance. Additionally, we investigate the influence of root multiplicity on the fractal structures of basin of attraction for equations with both simple and multiple roots.

Tipler naked singularities in N dimensions

Abstract A spacetime singularity, identified by the existence of incomplete causal geodesics in the spacetime, is called a (Tipler) strong curvature singularity if the volume form acting on independent Jacobi fields along causal geodesics vanishes in the approach of the singularity. It is called naked if at least one of these causal geodesics is past incomplete. Here, we study the formation of strong curvature naked singularities arising from spherically symmetric gravitational collapse of general type-I matter fields in an arbitrarily finite number of dimensions. In the spirit of Joshi and Dwivedi (1993 Phys. Rev. D 47 5357), and Goswami and Joshi (2007 Phys. Rev. D 76 084026), beginning with regular initial data, we derive two distinct (but not mutually exclusive) conditions, which we call the positive root condition (PRC) and the simple positive root condition (SPRC), that serve as necessary and sufficient conditions, respectively, for the existence of naked singularities. In doing so, we generalize the results of both the aforementioned works. We further restrict the PRC and the SPRC by imposing the curvature growth condition (CGC) of Clarke and Krolak (1985 J. Geom. Phys. 2(2) 127) on all causal curves that satisfy the causal convergence condition. The CGC then gives a sufficient condition ensuring that the naked singularities implying the PRC and implied by the SPRC, are of strong curvature type and hence correspond to the inextendibility of the spacetime. Using the CGC, we extend the results of Mosani et. al. (2020 Phys. Rev. D 101 044052) (that hold for dimension N=4) to the case N=5, showing that strong curvature naked singularities can occur in this case. However, for the case N≥6, we show that past-incomplete causal curves that identify naked singularities do not satisfy the CGC. These results shed light on the validity of the cosmic censorship conjectures in arbitrary dimensions.

Analytical solution of fractional oscillation equation with two Caputo fractional derivatives

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives , where the coefficients and orders satisfy and , is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that and are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large are determined as monotonous decays in a power of negative exponent.

Quadrature based innovative techniques concerning nonlinear equations having unknown multiplicity

Solution of nonlinear equations is one of the most frequently encountered issue in engineering and applied sciences. Most of the intricateed engineering problems are modeled in the frame work of nonlinear equation f(x)=0. The significance of iterative algorithms executed by computers in resolving such functions is of paramount importance and undeniable in contemporary times. If we study the simple roots and the roots having multiplicity greater of any nonlinear equations we come to the point that finding the roots of nonlinear equations having multiplicity greater than one is not trivialvia classical iterative methods. Instability or slow convergence rate is faced by these methods, and also sometimes these methods diverge. In this article, we give some innovative and robust iterative techniques for obtaining the approximate solution of nonlinear equations having multiplicity m>1. Quadrature formulas are implemented to obtain iterative techniques for finding roots of nonlinear equations having unknown multiplicity. The derived methods are the variants of modified Newton method with high order of convergence and better accuracy. The convergence criteria of the new techniques are studied by using Taylor series method. Some examples are tested for the sack of implementations of these techniques. Numerical and graphical comparison shows the performance and efficiency of these new techniques.

Biomechanical analysis of four different meniscus suturing techniques for posterior meniscal root pull-out repair: A human cadaveric study.

Tocompare the biomechanical properties of the slip-knot technique with three other transtibial pullout suture repair constructs for meniscal root tears. Thirty-two fresh-frozen cadaveric menisci were randomly allocated to four meniscus-suture fixation constructs: Two simple-sutures (TSS), two slip-knot (TSK) sutures, two cinch-loop (TCL) sutures, and two modified Mason-Allen (TMMA) sutures. Cyclic loading from 5 to 20 N was conducted for 1000 cycles at 0.5 Hz, and then loaded to failure at 0.5 mm/s. Parametric data (displacement during cyclic loading, ultimate load, yield load, and displacement at failure) were analysed using a one-way analysis of variance (ANOVA), whereas nonparametric data (stiffness) were analysed using the Kruskal-Wallis test. After 1000 cycles, the TCL construct significantly displaced the most (mean ± SD, 6.78 ± 1.32 mm; p < 0.001), followed by the TMMA (2.83 ± 0.90 mm), TSK (2.33 ± 0.57 mm), and TSS (2.03 ± 0.62 mm) groups. On ultimate failure load, there was no significant difference between the TSK group (123.48 ± 27.24 N, p > 0.05) and the other three groups (TSS, 94.65 ± 25.33 N; TMMA, 168.38 ± 23.24 N; TCL, 170.54 ± 57.32 N); however, it exhibited the least displacement (5.53 ± 1.25 mm) which was significantly shorter than those of the TCL (11.82 ± 4.25 mm, p < 0.001) and TMMA (9.53 ± 2.18 mm, p = 0.03) constructs. No significant difference in stiffness was observed among the four meniscus-suture constructs. The slip-knot technique has proven to be a simple, yet robust and stable meniscal root fixation option; moreover, it exhibited superiority over the more complex modified Mason-Allen suture construct in resisting displacement at the ultimate failure load. Not applicable.

Machine Learning Clifford Invariants of ADE Coxeter Elements

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_8$$\\end{document}, D8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_8$$\\end{document} and E8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_8$$\\end{document} for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.

Real univariate polynomials with given signs of coefficients and simple real roots

We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\ell ^+$ and $\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $\mathcal{P}^d:=\{ P:=x^d+a_1x^{d-1}+\dots +a_d\}$, $a_j\in \mathbb{R}^*=\mathbb{R}\setminus \{ 0\}$, the corresponding sign patterns $\sigma=(\sigma_1,\sigma_2,\dots, \sigma_d)$, where $\sigma_j=$sign$(a_j)$, and the sets $\mathcal{P}^d_{\sigma ,(\ell ^+,\ell ^-)}\subset \mathcal{P}^d$ of polynomials with given triples $(\sigma ,(\ell ^+,\ell ^-))$.We prove that for degree $d\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $\sigma =(-,-,-,+)$, $\ell^+=0$, $\ell ^-=2$, with $d=5$, $\sigma =(-,-,-,-,+)$, $\ell^+=0$, $\ell ^-=3$, and the ones obtained from them under the $\mathbb{Z}_2\times \mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting involutions: $i_m\colon P(x)\mapsto (-1)^dP(-x)$ and $i_r\colon P(x)\mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex conjugate pairs of roots ($2n\leq d$);2) for $1\leq s\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $d\geq 6$, we give an example of a set $\mathcal{P}^d_{\sigma ,(\ell^+,\ell^-)}$ having $\Lambda (d)$ connected compo\-nents, where $\Lambda (d)\rightarrow \infty$ as $d\rightarrow \infty$.

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

This research paper comprehensively presents the derivation of a seventh-order iteration scheme designed to obtain simple roots of nonlinear equations through the utilization of Lagrange interpolation technique. The scheme is characterized by the requirement for three function evaluations and one evaluation of the first derivative in each iteration. A detailed convergence analysis is also carried out to assess the efficacy of the proposed method. Additionally, the paper includes comprehensive numerical experiments aimed at confirming the theoretical results and illustrating the competitive performance of the derived iteration scheme.

Vertex operators for imaginary [formula omitted] subalgebras of the Monster Lie algebra

The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V♮⊗V1,1, where V♮ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and V1,1 is the vertex algebra corresponding to the rank 2 even unimodular lattice II1,1. We construct vertex algebra elements that project to bases for subalgebras of m isomorphic to gl2, corresponding to each imaginary simple root, denoted (1,j) for j>0. Our method requires the existence of pairs of primary vectors in V♮ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group M on the subspace of primary vectors in V♮ induces an M-action on the set of gl2 subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of V♮ to prove that this action is non-trivial for small values of j.

Application Of Multipoint Secant-Type Method ForFinding Roots 0f Nonlinear Equations

In this paper, we introduce a family of pk-order iterative schemes for finding the simple root of a nonlinear algebraic equation of the function fx=0 by using the divided difference approximation. The proposed method uses one evaluation of the function per iteration and can achieve convergence order pk. The error equation and asymptotic convergence constant are proved theoretically and numerically. Numerical examples are included to demonstrate the exceptional convergence speed of the proposed method and thus verify the theoretical results

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

Iterative method improving Newton's method with higher order

In the present article, we have constructed and looked at improving two- and three-step iterative methods to locate simple roots of non-linear equations. It appears that the three-step iterative method converges to the eighth order. The whole aim of this research is to derive and present a new modified iterative method of higher order and to obtain less iteration than the classical Newton method for solving nonlinear equations of simple roots. The present technique has to evaluate three functions and two first derivatives in each iteration. The advantage of the proposed method has been observed to have at least better performance effective and more stability by comparing with the other methods for the same or less than order. Also, noted that our method gives better results in terms of the number of iterations. To demonstrate the efficacy and popularity, different numerical illustrations are provided for the recommended techniques.

Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions

In this paper, we derive a spectral collocation method for solving fractional-order integro-differential equations by using a kind of Müntz orthogonal functions that are defined on 0,1 and have simple and real roots in this interval. To this end, we first construct the operator of Riemann–Liouville fractional integral corresponding to this kind of Müntz functions. Then, using the Gauss–Legendre quadrature rule and by employing the roots of Müntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. We also construct an upper bound for the truncation error of Müntz orthogonal functions, and we analyze the error of the proposed collocation method. Numerical examples are included to demonstrate the validity and accuracy of the method.